1 Introduction: The Iterative Modeling Process ................. 1
2 Modeling and Inverse Problems ................................ 7
2.1 Mechanical Vibrations ................................... 7
2.2 Inverse Problems ....................................... 11
References .................................................. 15
3 Mathematical and Statistical Aspects of Inverse Problems .... 17
3.1 Probability and Statistics Overview .................... 18
3.1.1 Probability ..................................... 18
3.1.2 Random Variables ................................ 20
3.1.3 Statistical Averages of Random Variables ........ 21
3.1.4 Special Probability Distributions ............... 22
3.2 Parameter Estimation or Inverse Problems ............... 29
3.2.1 The Mathematical Model .......................... 29
3.2.2 The Statistical Model ........................... 30
3.2.3 Known Error Processes: Maximum Likelihood
Estimators ...................................... 31
3.2.3.1 Normally Distributed Errors ............ 31
3.2.4 Unspecified Error Distributions and Asymptotic
Theory .......................................... 33
3.2.5 Ordinary Least Squares (OLS) .................... 34
3.2.6 Numerical Implementation of the Vector OLS
Procedure ....................................... 37
3.2.7 Generalized Least Squares (GLS) ................. 38
3.2.8 GLS Motivation .................................. 39
3.2.9 Numerical Implementation of the GLS Procedure ... 40
3.3 Computation of  n, Standard Errors and Confidence
Intervals .............................................. 41
3.4 Investigation of Statistical Assumptions ............... 45
3.4.1 Residual Plots .................................. 46
3.4.2 An Example Using Residual Plots ................. 47
3.5 Statistically Based Model Comparison Techniques ........ 51
3.5.1 RSS Based Statistical Tests ..................... 54
3.5.1.1 P-Values ............................... 56
3.5.1.2 Alternative Statement .................. 57
3.5.2 Application: Cat-Brain Diffusion/Convection
Problem .............................................. 57
References .................................................. 63
4 Mass Balance and Mass Transport ............................. 65
4.1 Introduction ........................................... 65
4.2 Compartmental Concepts ................................. 65
4.3 Compartment Modeling ................................... 67
4.4 General Mass Transport Equations ....................... 71
4.4.1 Mass Flux Law in a Stationary (Non-Moving)
Fluid ........................................... 73
4.4.2 Mass Flux in a Moving Fluid ..................... 75
References .................................................. 79
5 Heat Conduction ............................................. 81
5.1 Motivating Problems .................................... 81
5.1.1 Radio-Frequency Bonding of Adhesives ............ 81
5.1.2 Thermal Testing of Structures ................... 82
5.2 Mathematical Modeling of Heat Transfer ................. 83
5.2.1 Introduction .................................... 83
5.2.2 Fourier's Law of Heat Conduction ................ 84
5.2.3 Heat Equation ................................... 85
5.2.4 Boundary Conditions and Initial Conditions ...... 89
5.2.5 Properties of Solutions ......................... 93
5.3 Experimental Modeling of Heat Transfer ................. 94
5.3.1 The Thermocouple as a Temperature Measuring
Device .......................................... 95
5.3.2 Detailed Hardware and Software Lists ............ 98
References ................................................. 101
6 Structural Modeling: Force/Moments Balance ................. 103
6.1 Motivation: Control of Acoustics/Structural
Interactions .......................................... 103
6.2 Introduction to Mechanics of Elastic Solids ........... 104
6.2.1 Normal Stress and Strain ....................... 105
6.2.2 Stress and Strain Relationship (Hooke's Law) ... 106
6.2.3 Shear Stress and Strain ........................ 109
6.3 Deformations of Beams ................................. 112
6.3.1 Differential Equations of Thin Beam
Deflections .................................... 114
6.3.1.1 Force Balance ......................... 114
6.3.1.2 Moment Balance ........................ 116
6.3.1.3 Moment Computation .................... 117
6.3.1.4 Initial Conditions .................... 123
6.3.1.5 Boundary Conditions ................... 124
6.4 Separation of Variables: Modes and Mode Shapes ........ 129
6.5 Numerical Approximations: Galerkin's Method ........... 135
6.6 Energy Functional Formulation ......................... 141
6.7 The Finite Element Method ............................. 143
6.8 Experimental Beam Vibration Analysis .................. 147
References ................................................. 153
7 Beam Vibrational Control and Real-Time Implementation ...... 155
7.1 Introduction .......................................... 155
7.2 Controllability and Observability of Linear Systems ... 155
7.2.1 Controllability ................................ 156
7.2.1.1 Time-Varying Case ..................... 156
7.2.1.2 Time-Invariant Case ................... 162
7.2.2 Observability .................................. 170
7.2.2.1 Time-Varying Case ..................... 170
7.2.2.2 Time-Invariant Case ................... 172
7.3 Design of State Feedback Control Systems and State
Estimators ............................................ 175
7.3.1 Effect of State Feedback on System
Properties ..................................... 179
7.3.1.1 Stability ............................. 179
7.3.1.2 Controllability ....................... 180
7.3.1.3 Observability ......................... 181
7.4 Pole Placement (Relocation) Problem ................... 182
7.4.1 State Estimator (Luenberger Observer) .......... 190
7.4.2 Dynamic Output Feedback Compensator ............ 191
7.5 Linear Quadratic Regulator Theory ..................... 197
7.6 Beam Vibrational Control: Real-Time Feedback Control
Implementation ........................................ 200
References ................................................. 213
8 Wave Propagation ........................................... 215
8.1 Fluid Dynamics ........................................ 215
8.1.1 Newton's Law of Viscosity ...................... 216
8.1.2 Derivative in Fluid Flows ...................... 220
8.1.3 Equations of Fluid Motion ...................... 220
8.2 Fluid Waves ........................................... 229
8.2.1 Terminology .................................... 229
8.2.2 SoundWaves ..................................... 231
8.2.2.1 Euler's Equation ...................... 233
8.2.2.2 Equation of Continuity ................ 233
8.2.2.3 Equation of State ..................... 233
8.2.3 Wave Equations ................................. 234
8.3 Experimental Modeling of the Wave Equation ............ 238
References ................................................. 243
9 Size-Structured Population Models .......................... 245
9.1 Introduction: A Motivating Application ................ 245
9.2 A Single Species Model (Malthusian Law) ............... 246
9.3 The Logistic Model .................................... 247
9.4 A Predator/Prey Model ................................. 249
9.5 A Size-Structured Population Model .................... 251
9.6 The Sinko-Streifer Model and Inverse Problems ......... 265
9.7 Size Structure and Mosquitofish Populations ........... 268
References ................................................. 277
A An Introduction to Fourier Techniques ...................... 281
A.l Fourier Series ........................................ 281
A.2 Fourier Transforms .................................... 284
В Review of Vector Calculus .................................. 287
References ................................................. 293
Index ......................................................... 295
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